To solve this inequality, we need to first simplify both sides of the inequality:
lg(x^2 - 6) > lg(2x - 3)
Next, we can rewrite the logarithmic inequality in exponential form:
x^2 - 6 > 2x - 3
Now, let's solve for x:
x^2 - 2x + 3 > 0
(x - 3)(x - 1) > 0
The critical points are x = 1 and x = 3. We can now test the intervals created by these critical points:
When x < 1:Both factors are negative, resulting in a positive product. This interval satisfies the inequality.
When 1 < x < 3:One factor is positive and one factor is negative, resulting in a negative product. This interval does not satisfy the inequality.
When x > 3:Both factors are positive, resulting in a positive product. This interval satisfies the inequality.
Therefore, the solution to the inequality lg(x^2-6) > lg(2x-3) is x < 1 or x > 3.
To solve this inequality, we need to first simplify both sides of the inequality:
lg(x^2 - 6) > lg(2x - 3)
Next, we can rewrite the logarithmic inequality in exponential form:
x^2 - 6 > 2x - 3
Now, let's solve for x:
x^2 - 2x + 3 > 0
(x - 3)(x - 1) > 0
The critical points are x = 1 and x = 3. We can now test the intervals created by these critical points:
When x < 1:
Both factors are negative, resulting in a positive product. This interval satisfies the inequality.
When 1 < x < 3:
One factor is positive and one factor is negative, resulting in a negative product. This interval does not satisfy the inequality.
When x > 3:
Both factors are positive, resulting in a positive product. This interval satisfies the inequality.
Therefore, the solution to the inequality lg(x^2-6) > lg(2x-3) is x < 1 or x > 3.